Theorem iccss 11592

Description: Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 20-Feb-2015.)

Assertion

Ref	Expression
iccss	|- ( ( ( A e. RR /\ B e. RR ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) )

Proof of Theorem iccss

Dummy variables x w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexr 9639	. . 3 |- ( A e. RR -> A e. RR* )
2		rexr 9639	. . 3 |- ( B e. RR -> B e. RR* )
3	1, 2	anim12i 566	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A e. RR* /\ B e. RR* ) )
4		df-icc 11536	. . 3 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
5		xrletr 11361	. . 3 |- ( ( A e. RR* /\ C e. RR* /\ w e. RR* ) -> ( ( A <_ C /\ C <_ w ) -> A <_ w ) )
6		xrletr 11361	. . 3 |- ( ( w e. RR* /\ D e. RR* /\ B e. RR* ) -> ( ( w <_ D /\ D <_ B ) -> w <_ B ) )
7	4, 4, 5, 6	ixxss12 11549	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) )
8	3, 7	sylan 471	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A <_ C /\ D <_ B ) ) -> ( C [,] D ) C_ ( A [,] B ) )

Syntax hints:   -> wi 4   /\ wa 369   e. wcel 1767   C_ wss 3476   class class class wbr 4447  (class class class)co 6284   RRcr 9491   RR*cxr 9627   <_ cle 9629   [,]cicc 11532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-pre-lttri 9566  ax-pre-lttrn 9567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-icc 11536

This theorem is referenced by:  xrhmeo  21209  lebnumii  21229  pcoval1  21276  pcoval2  21279  ivthicc  21633  dyaddisjlem  21767  volsup2  21777  volcn  21778  mbfi1fseqlem5  21889  dvcvx  22184  dvfsumle  22185  dvfsumabs  22187  harmonicbnd3  23093  ppisval  23133  chtwordi  23186  ppiwordi  23192  chpub  23251  cvmliftlem2  28399  fourierdlem76  31511  fourierdlem103  31538  fourierdlem104  31539  fourierdlem107  31542  fourierdlem112  31547